Copied to
clipboard

## G = C22.D36order 288 = 25·32

### 1st non-split extension by C22 of D36 acting via D36/D18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C22.D36
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×C18 — C2×C9⋊D4 — C22.D36
 Lower central C9 — C18 — C2×C18 — C22.D36
 Upper central C1 — C2 — C23 — C22⋊C4

Generators and relations for C22.D36
G = < a,b,c,d | a2=b2=c36=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >

Subgroups: 420 in 78 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22 [×3], C22 [×3], S3, C6, C6 [×3], C2×C4 [×3], D4 [×2], C23, C23, C9, Dic3 [×2], C12, D6 [×2], C2×C6 [×3], C2×C6, C22⋊C4, C22⋊C4, C2×D4, D9, C18, C18 [×3], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C22×S3, C22×C6, C23⋊C4, Dic9 [×2], C36, D18 [×2], C2×C18 [×3], C2×C18, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C2×Dic9, C2×Dic9, C9⋊D4 [×2], C2×C36, C22×D9, C22×C18, C23.6D6, C18.D4, C9×C22⋊C4, C2×C9⋊D4, C22.D36
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C23⋊C4, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C23.6D6, D18⋊C4, C22.D36

Smallest permutation representation of C22.D36
On 72 points
Generators in S72
```(1 54)(3 56)(5 58)(7 60)(9 62)(11 64)(13 66)(15 68)(17 70)(19 72)(21 38)(23 40)(25 42)(27 44)(29 46)(31 48)(33 50)(35 52)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 44 54 27)(2 43)(3 25 56 42)(4 24)(5 40 58 23)(6 39)(7 21 60 38)(8 20)(9 72 62 19)(10 71)(11 17 64 70)(12 16)(13 68 66 15)(14 67)(18 63)(22 59)(26 55)(28 36)(29 52 46 35)(30 51)(31 33 48 50)(34 47)(37 61)(41 57)(45 53)(65 69)```

`G:=sub<Sym(72)| (1,54)(3,56)(5,58)(7,60)(9,62)(11,64)(13,66)(15,68)(17,70)(19,72)(21,38)(23,40)(25,42)(27,44)(29,46)(31,48)(33,50)(35,52), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,44,54,27)(2,43)(3,25,56,42)(4,24)(5,40,58,23)(6,39)(7,21,60,38)(8,20)(9,72,62,19)(10,71)(11,17,64,70)(12,16)(13,68,66,15)(14,67)(18,63)(22,59)(26,55)(28,36)(29,52,46,35)(30,51)(31,33,48,50)(34,47)(37,61)(41,57)(45,53)(65,69)>;`

`G:=Group( (1,54)(3,56)(5,58)(7,60)(9,62)(11,64)(13,66)(15,68)(17,70)(19,72)(21,38)(23,40)(25,42)(27,44)(29,46)(31,48)(33,50)(35,52), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,44,54,27)(2,43)(3,25,56,42)(4,24)(5,40,58,23)(6,39)(7,21,60,38)(8,20)(9,72,62,19)(10,71)(11,17,64,70)(12,16)(13,68,66,15)(14,67)(18,63)(22,59)(26,55)(28,36)(29,52,46,35)(30,51)(31,33,48,50)(34,47)(37,61)(41,57)(45,53)(65,69) );`

`G=PermutationGroup([(1,54),(3,56),(5,58),(7,60),(9,62),(11,64),(13,66),(15,68),(17,70),(19,72),(21,38),(23,40),(25,42),(27,44),(29,46),(31,48),(33,50),(35,52)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,44,54,27),(2,43),(3,25,56,42),(4,24),(5,40,58,23),(6,39),(7,21,60,38),(8,20),(9,72,62,19),(10,71),(11,17,64,70),(12,16),(13,68,66,15),(14,67),(18,63),(22,59),(26,55),(28,36),(29,52,46,35),(30,51),(31,33,48,50),(34,47),(37,61),(41,57),(45,53),(65,69)])`

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 18J ··· 18O 36A ··· 36L order 1 2 2 2 2 2 3 4 4 4 4 4 6 6 6 6 6 9 9 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 2 2 36 2 4 4 36 36 36 2 2 2 4 4 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 S3 D4 D6 D9 C4×S3 D12 C3⋊D4 D18 C4×D9 D36 C9⋊D4 C23⋊C4 C23.6D6 C22.D36 kernel C22.D36 C18.D4 C9×C22⋊C4 C2×C9⋊D4 C2×Dic9 C22×D9 C3×C22⋊C4 C2×C18 C22×C6 C22⋊C4 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C9 C3 C1 # reps 1 1 1 1 2 2 1 2 1 3 2 2 2 3 6 6 6 1 2 6

Matrix representation of C22.D36 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 36 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 0 0 20 31 0 0 6 26 2 26 0 0 11 13 0 0
,
 24 26 0 0 2 13 0 0 0 0 26 6 0 0 17 11
`G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[0,0,2,11,0,0,26,13,20,6,0,0,31,26,0,0],[24,2,0,0,26,13,0,0,0,0,26,17,0,0,6,11] >;`

C22.D36 in GAP, Magma, Sage, TeX

`C_2^2.D_{36}`
`% in TeX`

`G:=Group("C2^2.D36");`
`// GroupNames label`

`G:=SmallGroup(288,13);`
`// by ID`

`G=gap.SmallGroup(288,13);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^2=c^36=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;`
`// generators/relations`

׿
×
𝔽